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Factorial
In mathematics, the factorial of a positive integer n , denoted by n! , is the product of all positive integers less than or equal to . For example, : 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 500 \,. The value of 0! is 1, according to the convention for an empty product. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. Its most basic use counts the possible distinct sequences -- the permutations -- of n distinct objects: there are n! . The factorial function can also be extended to non-integer arguments while retaining its most important properties; this involves more advanced mathematics, notably techniques from mathematical analysis. Table of values These are selected members of the factorial sequence, the values > 20! are written in scientific notation and rounded to 10 significant digits. Definition The factorial function is defined by the product : n! = 1 \cdot 2 \cdot 3 \cdots (n-2) \cdot (n-1) \cdot n, for integer . This may be written in the Pi product notation as : n! = \prod_{i = 1}^n i. From these formulas, one may derive the recurrence relation : n! = n \cdot (n-1)! . For example, one has : \begin{align} 4! &= 4 \cdot 3! \\ 6! &= 6 \cdot 5! \\ 100! &= 100 \cdot \mathcal{E}\mathcal{E}! \end{align} and so on. Factorial of zero The factorial of 0, 0! , is 1. There are several motivations for this definition: * For , the definition of as a product involves the product of no numbers at all, and so is an example of the broader convention that the product of no factors is equal to the multiplicative identity (see empty product). * There is exactly one permutation of zero objects (with nothing to permute, the only rearrangement is to do nothing). * It makes many identities in combinatorics valid for all applicable sizes. The number of ways to choose 0 elements from the empty set is given by the binomial coefficient :: \binom{0}{0} = \frac{0!}{0!0!} = 1 . : More generally, the number of ways to choose all elements among a set of is :: \binom{n}{n} = \frac{n!}{n!0!} = 1 . * It allows for the compact expression of many formulae, such as the exponential function, as a power series: :: e^x = \sum_{n = 0}^\infty \frac{x^n}{n!}. * It extends the recurrence relation to 0. Factorial of a non-integer The factorial function can also be defined for non-integer values using more advanced mathematics (the gamma function ), detailed in the section below. This more generalized definition is used by advanced calculators and mathematical software such as Maple, Mathematica, or APL. Applications Although the factorial function has its roots in combinatorics, formulas involving factorials occur in many areas of mathematics. * There are different ways of arranging distinct objects into a sequence, the permutations of those objects. * Often factorials appear in the denominator of a formula to account for the fact that ordering is to be ignored. A classical example is counting -combinations (subsets of elements) from a set with elements. One can obtain such a combination by choosing a -permutation: successively selecting and removing one element of the set, times, for a total of :: (n-0)(n-1)(n-2)\cdots\left(n-(k-1)\right) = \tfrac{n!}{(n-k)!} = n^{\underline k} :possibilities. This however produces the -combinations in a particular order that one wishes to ignore; since each -combination is obtained in different ways, the correct number of -combinations is :: \frac{n(n-1)(n-2)\cdots(n-k+1)}{k(k-1)(k-2)\cdots 1} = \frac{n^{\underline k}}{k!}= \frac{n!}{(n-k)!k!} = \binom {n}{k}. :This number is known as the binomial coefficient, because it is also the coefficient of in . The term n^{\underline k} is often called a falling factorial (pronounced "n'' to the falling ''k"). * Factorials occur in algebra for various reasons, such as via the already mentioned coefficients of the binomial formula, or through averaging over permutations for symmetrization of certain operations. * Factorials also turn up in calculus; for example, they occur in the denominators of the terms of Taylor's formula, where they are used as compensation terms due to the th derivative of being equivalent to . * Factorials are also used extensively in probability theory and number theory (see below). * Factorials can be useful to facilitate expression manipulation. For instance the number of -permutations of can be written as :: n^{\underline k}=\frac{n!}{(n-k)!}\,; :while this is inefficient as a means to compute that number, it may serve to prove a symmetry property of binomial coefficients: :: \binom nk=\frac{n^{\underline k}}{k!}=\frac{n!}{(n-k)!k!} = \frac{n^{\underline{n-k}}}{(n-k)!} = \binom n{n-k}\,. * The factorial function can be shown, using the power rule, to be :: n! = D^n\,x^n = \frac{d^n}{dx^n}\,x^n :where is Euler's notation for the th derivative of . Category:Pages